On Automorphism Groups of Finite Chain Rings

Abstract

A finite ring with an identity is a chain ring if its lattice of left ideals forms a unique chain. Let R be a finite chain ring with invaraints p,n,r,k,k′,m. If n=1, the automorphism group Aut(R) of R is known. The main purpose of this article is to study the structure of Aut(R) when n>1. First, we prove that Aut(R) is determined by the automorphism group of a certain commutative chain subring. Then we use this fact to find the automorphism group of R when p∤k. In addition, Aut(R) is investigated under a more general condition; that is, R is very pure and p need not divide k. Based on the j-diagram introduced by Ayoub, we were able to give the automorphism group in terms of a particular group of matrices. The structure of the automorphism group of a finite chain ring depends essentially on its invaraints and the associated j-diagram.

Keywords

• finite chain ring
• automorphism group
• Galois ring
• j-diagram